A new partial ordering of knots

Abstract

Our research concerns how knots behave under crossing changes. In particular, we investigate a partial ordering of alternating knots that results from performing crossing changes. A similar ordering was originally introduced by Kouki Taniyama in the paper “A partial order of knots”. We amend Taniyama’s partial ordering and present theorems about the structure of our ordering for more complicated knots. Our approach is largely graph theoretic, as we translate each knot diagram into one of two planar graphs by checkerboard coloring the plane. Of particular interest are the class of knots known as pretzel knots, as well as knots that have only one direct minor in the partial ordering.